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A296456
Decimal expansion of limiting power-ratio for A296266; see Comments.
1
2, 6, 3, 8, 7, 6, 2, 8, 9, 9, 1, 3, 8, 5, 1, 1, 7, 7, 5, 3, 8, 3, 3, 2, 0, 7, 8, 1, 2, 3, 2, 0, 7, 3, 6, 6, 5, 0, 3, 0, 3, 9, 3, 2, 0, 1, 8, 0, 4, 5, 5, 2, 4, 4, 6, 6, 5, 6, 4, 2, 7, 1, 8, 5, 2, 5, 0, 7, 9, 7, 3, 0, 4, 8, 2, 8, 9, 5, 0, 3, 0, 7, 8, 7, 2, 4
OFFSET
2,1
COMMENTS
Suppose that A = {a(n)}, for n >= 0, is a sequence, and g is a real number such that a(n)/a(n-1) -> g. The limiting power-ratio for A is the limit as n->oo of a(n)/g^n, assuming that this limit exists. For A = A296266 we have g = (1 + sqrt(5))/2, the golden ratio (A001622). See A296425-A296434 for related ratio-sums and A296452-A296461 for related limiting power-ratios.
EXAMPLE
Limiting power-ratio = 26.38762899138511775383320781232073665030...
MATHEMATICA
a[0] = 1; a[1] = 2; b[0] = 3; b[1] = 4; b[2] = 5;
a[n_] := a[n] = a[n - 1] + a[n - 2] + b[n]*b[n - 2];
j = 1; While[j < 12, k = a[j] - j - 1;
While[k < a[j + 1] - j + 1, b[k] = j + k + 2; k++]; j++];
Table[a[n], {n, 0, 15}] (* A296266 *)
z = 2000; g = GoldenRatio; h = Table[N[a[n]/g^n, z], {n, 0, z}];
StringJoin[StringTake[ToString[h[[z]]], 41], "..."]
Take[RealDigits[Last[h], 10][[1]], 120] (* A296456 *)
CROSSREFS
Sequence in context: A076041 A156816 A021383 * A256592 A256965 A191708
KEYWORD
nonn,easy,cons
AUTHOR
Clark Kimberling, Dec 15 2017
STATUS
approved